You may want to keep the original version of Tinycore, as well as install the more recent version.

To do this, using a root file manager, create a new directory in /mnt/sda1 (use the correct partition if not sda1) for the more recent version. Let's call the new directory tc3.5.1 when installing tinycore 3.5.1, and call the new directory tc3.6 when installing tinycore 3.6.

Paste bzImage and tinycore.gz to the new directory you created.

Update the bootloader.

If using Extlinux, using a root text editor (sudo editor), open /mnt/sda1/boot/extlinux/extlinux.conf